Given the constraint that the sum of the number and its reversed form is 888 and focusing on the clue that the units digit in the sum scenario must add up to 8, we look directly at the options provided:

– (a) 339: The sum of 3 and 9 does not lead to an end digit of 8 in the sum.
– (b) 341: The sum of 1 and 4 does not lead to an end digit of 8 in the sum.
– (c) 378: The sum of 8 and 7 does not directly address the end digit sum condition.
– (d) 345: When 345 is added to its reverse 543, the unit digits 5 and 3 indeed add up to 8, meeting the immediate condition.

The specific insight about the unit’s digits adding up to 8 being satisfied only by option (d) “345” (since \(3+5=8\)) simplifies the approach significantly. However, to validate this option fully in the context of the entire problem:

– If the original number is 345 and its reverse is 543, their sum is indeed 888 (\(345 + 543 = 888\)), which satisfies one of the problem’s conditions.
– The second condition mentioned is that swapping the unit’s and ten’s digit of the original number results in a number that is 9 more than the original. Swapping the units and tens digit of 345 gives 354, which is indeed 9 more than 345 (\(354 – 345 = 9\)).

Therefore, considering both conditions and the insight provided about the sum leading to the last digits adding up to 8, the correct number is indeed option (d) **345**. This choice fulfills both specified conditions of the problem: the sum with its reversed form equals 888, and swapping the tens and units digits results in a number that is 9 more than the original.

Given the operation defined by \(p \times q = p + q + \frac{p}{q}\), let’s calculate the value of \(8 \times 2\) using this operation.

Calculating \(8 \times 2\) Using the Given Operation

Substitute \(p = 8\) and \(q = 2\) into the given formula:

\[
8 \times 2 = 8 + 2 + \frac{8}{2}
\]

Simplify the expression:

\[
8 \times 2 = 10 + 4 = 14
\]

Therefore, according to the given operation, \(8 \times 2 = \boldsymbol{14}\).

Identification of the Unique Bus Number

The task is to identify a bus number within the range of 1 to 500 that is not only a perfect square but also remains a perfect square when the digits are viewed upside down. Following the clarified criteria:

Criteria for the Bus Number

• The number must be a perfect square.
• When turned upside down, the number must still represent a perfect square.
• The number must include only the digits 0, 1, 6, 8, and 9, as these are the digits that can represent other numbers or themselves when flipped.

Analysis

We consider the range of perfect squares from \(1^2\) to \(22^2\) as these are the perfect squares within the 1 to 500 range:

\[
1, 4, 9, \ldots, 484\ (22^2)
\]

Among these, the number that meets the specific requirement of being a perfect square that, when turned upside down, also becomes a perfect square, is 169:

– Original Number: \(169 = 13^2\), a perfect square.
– Upside Down: When 169 is flipped upside down, it becomes 961. Notably, 6 and 9 flip, while 1 remains the same, making the new figure, 961, which is \(31^2\), another perfect square.

Conclusion

Therefore, the bus number with the described peculiarity is 169. This number is unique in that it satisfies the condition of being a perfect square and also transforms into another perfect square (961) when viewed upside down.

Let’s denote the number of buffaloes as \(b\) and the number of ducks as \(d\).

Since each buffalo has 4 legs and each duck has 2 legs, the total number of legs in the group is \(4b + 2d\). The total number of heads, which is also the total number of animals, is \(b + d\).

According to the problem, the number of legs is 24 more than twice the number of heads. Therefore, we can write the equation:

\[4b + 2d = 2(b + d) + 24\]

Simplifying this equation:

\[4b + 2d = 2b + 2d + 24\]
\[2b = 24\]
\[b = 12\]

So, there are 12 buffaloes in the group.

Given the equation:
\[7 \sin^2 \theta + 3 \cos^2 \theta = 4\]

Since \(\cos^2 \theta = 1 – \sin^2 \theta\), we can substitute this into the equation:
\[7 \sin^2 \theta + 3 (1 – \sin^2 \theta) = 4\]

Expanding and simplifying:
\[7 \sin^2 \theta + 3 – 3 \sin^2 \theta = 4\]
\[4 \sin^2 \theta = 1\]
\[\sin^2 \theta = \frac{1}{4}\]
\[\sin \theta = \pm \frac{1}{2}\]

Since \(0^\circ < \theta < 90^\circ\), we know that \(\sin \theta\) is positive in this interval. Therefore, we can disregard the negative solution, leaving us with:
\[\sin \theta = \frac{1}{2}\]

The value of \(\theta\) that satisfies this equation in the given interval is:
\[\theta = 30^\circ \text{ or } \frac{\pi}{6} \text{ radians}\]

So, the value of \(\theta\) is \(30^\circ\) or \(\frac{\pi}{6}\) radians.